Waveform shapes for treating neurological disorders optimized for energy efficiency

ABSTRACT

Systems and methods for stimulation of neurological tissue apply a stimulation waveform that is derived by a developed genetic algorithm (GA), which may be coupled to a computational model of extracellular stimulation of a mammalian myelinated axon. The waveform is optimized for energy efficiency.

RELATED APPLICATION

This application claims the benefit of U.S. Provisional PatentApplication Ser. No. 61/348,963, filed May 27, 2010, and entitled“Energy-Optimal Biphasic Waveform Shapes for Neural Stimulation,” whichis incorporated herein by reference.

GOVERNMENT LICENSING RIGHTS

This invention was made in part with government support under NIH GrantNos. R01 NS040894 and R21 NS054048. The government has certain rights tothe invention.

FIELD OF THE INVENTION

This invention relates to systems and methods for stimulating nerves inmammals and, in particular, humans.

BACKGROUND OF THE INVENTION

Implantable and external electrical stimulators assist thousands ofindividuals with neurological disorders. These stimulators generateelectrical waveforms, which are delivered by leads to targeted tissueregions to treat the neurological disorders. Examples of treatingneurological disorders using electrical stimulation include deep brainstimulation, cortical stimulation, vagus nerve stimulation, sacral nervestimulation, spinal cord stimulation, and cardiac pacemakers anddefibrillators.

Implantable stimulators are powered by either primary cell orrechargeable batteries. When the energy of a primary cell battery isdepleted, the entire stimulator must be replaced through an expensiveand invasive surgical procedure. The energy capacity of a rechargeablebattery determines the recharge interval, as well as the overall volumeof the implant.

There are clinical benefits to reducing the frequency ofbattery-replacement surgeries or recharge intervals, as well as reducingthe physical size (volume) of the stimulator itself. The problem is howone alters stimulation parameters to achieve this objective withoutsacrificing clinical efficacy and generating unwanted side effects. Forexample, the energy efficiency of stimulation (i.e., how much energy isconsumed for the generation of a given stimulation pulse) cannot beviewed in isolation. The charge efficiency of stimulation is also animportant consideration with implanted devices. The charge deliveredduring a stimulus pulse contributes to the risk of tissue damage (Yuenet al. 1981; McCreery et al. 1990). If energy-efficient stimulationparameters deliver excessive amounts of charge, then the benefits ofhigh energy efficiency are diminished.

As shown in FIGS. 1A and 1B, the energy efficiency of stimulationparameters is dependent upon the amplitude of the stimulation pulse(typically expressed, e.g., in a range from 10 μA upwards to 10 mA); thewidth or duration of the stimulation pulse (typically expressed, e.g.,in a range from 20 μs upwards to 500 μs); the frequency of the pulsesapplied over time (typically expressed, e.g., in a range from 10 Hzupwards to 200 Hz); and the shape or waveform of the pulse (e.g.,typically, depending upon the therapeutic objective, square(rectangular) (see FIG. 2A), or rising ramp (see FIG. 2B), or sinusoid(see FIG. 2C), or decreasing exponential (see FIG. 2D), or risingexponential (see FIG. 2E)).

Previous studies have used passive membrane models to analyze theeffects of waveform shape on efficiency. All previous studies usingpassive membrane models have concluded that the energy-optimal waveformshape is a rising exponential (Offner 1946; Fishler 2000; Kajimoto etal. 2004; Jezernik and Morari 2005).

However, in more realistic models and in vivo experiments, the inventorshave found that the rising exponential waveform proved to be no moreenergy-efficient than rectangular, ramp, or decaying exponentialwaveforms. In fact, in realistic membrane models, the inventors havefound that energy-optimal waveform shapes cannot be determinedanalytically because of the complexity and non-linearity of theequations that define the excitable membrane in the model. Also, a“brute force” method of testing every possible waveform shape is notfeasible since the number of possible waveform shapes is infinite.

SUMMARY OF THE INVENTION

One aspect of the invention provides systems and methodologies thatcouple an optimization algorithm, such as a global optimizationalgorithm (e.g. a genetic algorithm) to a computational model ofextracellular stimulation of a mammalian myelinated axon, to derive aset of stimulus waveforms that are optimized for a desired parameter,such as energy efficiency.

One aspect of the invention provides systems and methodologies thatcouple a genetic algorithm (GA) to a computational model ofextracellular stimulation of a mammalian myelinated axon, to derive aset of stimulus waveforms that are optimized for energy efficiency. Thisaspect of the invention makes it possible in a systematic way togenerate and analytically validate energy-optimal waveform shapes.

Another aspect of the invention provides systems and methodologies thatinclude a set of stimulation waveforms that are optimized using aspecially configured genetic algorithm (GA) to be more energy-efficientthan conventional waveforms used in neural stimulation, as well as moreenergy-efficient than the conventional waveforms for excitation of nervefibers in vivo. The optimized GA waveforms are also charge-efficient.

The optimized energy-efficiency of the stimulation waveforms derivedaccording to the invention make it possible to prolong battery life ofstimulators, thus reducing the frequency of recharge intervals, thecosts and risks of battery-replacement surgeries, and the volume ofimplantable stimulators.

The set of stimulus waveforms optimized according to the invention forenergy efficiency can be readily applied to deep brain stimulation, totreat a variety of neurological disorders, such as Parkinson's disease,movement disorders, epilepsy, and psychiatric disorders such asobsessive-compulsion disorder and depression, and other indications,such as tinnitus. The set of stimulus waveforms optimized according tothe invention for energy efficiency can also be readily applied to otherclasses of electrical stimulation of the nervous system including, butnot limited to, cortical stimulation, and spinal cord stimulation, toprovide the attendant benefits described above and to treat diseases orindications such as but not limited to Parkinson's Disease, EssentialTremor, Movement Disorders, Dystonia, Epilepsy, Pain, Tinnitus,psychiatric disorders such as Obsessive Compulsive Disorder, Depression,and Tourette's Syndrome.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a first diagrammatic view (amplitude vs. time) of astimulation waveform indicating stimulation parameters of a hypotheticalneural stimulation train.

FIG. 1B is a second diagrammatic view (power vs. time) of a stimulationwaveform indicating stimulation parameters of a hypothetical neuralstimulation train.

FIGS. 2A to 2E are diagrammatic views of typical waveforms used forneural stimulation.

FIG. 3 is an anatomic view of a system for stimulating tissue of thecentral nervous system that includes a lead implanted in brain tissuecoupled to a pulse generator that is programmed with stimulationparameters to provide a stimulus waveform that has been optimized forenergy efficiency by coupling a genetic algorithm (GA) to acomputational model of extracellular stimulation of a mammalianmyelinated axon.

FIGS. 4A to 4C are flow charts diagrammatically showing the operation ofthe genetic algorithm (GA) coupled to the computational model ofextracellular stimulation of a mammalian myelinated axon.

FIGS. 5A and 5B are diagrammatic views of the computational model ofextracellular stimulation of a mammalian myelinated axon, which iscoupled to the genetic algorithm (GA).

FIGS. 6A and 6B illustrate the progression of the genetic algorithm (GA)coupled to the computational model of extracellular stimulation of amammalian myelinated axon for a single trial (stimulation pulse width(PW)=0.5 ms), FIG. 6A being a sequence of plots showing changes inwaveform shapes across generations and the most energy-efficientwaveform at each indicated generation, and FIG. 6B being a graph showingthe minimum and mean energy of population across 10,000 generations, asconvergence toward a common optimal energy-efficiency value occurs.

FIG. 7 shows curves of the energy-optimal stimulation waveformsresulting from the GA coupled to the computational model ofextracellular stimulation of a mammalian myelinated axon, for differentPWs, the curves representing the means of the resulting waveforms acrossfive independent trials, and the gray regions defining 95% confidenceintervals, the waveforms for PW=1 and 2 ms being combined, and theleading and trailing tails of low amplitude were truncated.

FIG. 8 is a representative input/output (I/O) curve that was constructedwhen evaluating the energy-efficiency of the GA waveforms in apopulation model of one hundred (100) parallel MRG axons (11.5-μmdiameter) distributed uniformly within a cylinder with 3-mm diameter.

FIGS. 9A to 9C are plots showing the energy efficiency of the GAwaveforms in a model of extracellular stimulation of a population ofmyelinated axons, FIG. 9A showing the energy-duration curves foractivation of 500 of the axons (mean+/−SE; n=10 different randompopulations of 100 axons), FIG. 9B showing the energy efficiency of theGA waveforms compared to conventional waveform shapes used in neuralstimulation (mean, n=10; SE was negligible) (positive values of “%difference with GA waveform” indicate that the GA waveforms were moreenergy-efficient) and FIG. 9C showing the energy efficiency plottedagainst charge efficiency.

FIGS. 10A, 10B, and 10C are sensitivity plots for the GA waveforms tomodel parameters, FIG. 10A showing sensitivity to fiber diameter (D)(curves represent mean of the GA waveforms across 5 trials for PW=0.1ms), and FIGS. 10B and 10C showing sensitivity to the Hodgkin-Huxleymodel (skewed Gaussian curves resulted) (curves represent the means ofthe resulting waveforms across 5 independent trials, and the grayregions define 95% confidence intervals for PW=0.2 ms (b) and PW=0.02 ms(c)) (amplitudes are not to scale). Additionally, the GA waveforms wereshown to be insensitive to the number of waveforms per generationpopulation, the number of surviving waveforms per generation, theaverage initial amplitude of the waveforms, and the mutation rate. TheGA waveforms were shown to be sensitive to changes in dt (smaller dtleads to more energy-efficient for short PW, and less energy-efficientfor long PW).

FIGS. 11A and 11B show the set up for the in vivo evaluation of the GAwaveforms.

FIGS. 12A, 12B, and 12C show the in vivo measurements of energyefficiency of neural stimulation with the GA waveforms, FIG. 12A showingthe energy-duration curves for generation of 50% of maximal EMG(mean+/−SE; n=3), FIG. 12B showing the energy efficiency of GA waveformscompared to rectangular and decaying exponential waveforms (mean+/−SE;n=3) (positive values of “% difference with GA waveform” indicate thatGA waveforms were more energy-efficient), and FIG. 12C showing energyefficiency plotted against charge efficiency.

FIG. 13 shows the energy-optimal biphasic GA waveforms resulting fromthe biphasic GA waveforms for varying duration and timing of the anodicphase (the curves represent the mean of the cathodic phases of thewaveforms across 5 trials of the GA, and waveforms were shifted to alignthe peaks).

FIGS. 14A to 14H show the energy efficiency of biphasic GA waveforms ina model of extracellular stimulation of a population of myelinatedaxons, FIGS. 14A and 14B being energy-duration curves for activation of50% of the axons (mean+/−SE; n=5 different random populations of 100axons), FIGS. 14C to 14H being energy efficiency of GA waveformscompared to conventional waveform shapes used in neural stimulation(mean+/−SE, n=5) (positive values of “% difference with GA waveform”indicate that GA waveforms were more energy-efficient), the Figuresshowing that waveforms with cathodic phase first were moreenergy-efficient than waveforms with anodic phase first forPW_(cathodic)≦0.2 ms, 0.05 ms, and 0.05 ms forPW_(anodic)/PW_(cathodic)=1, 5, and 10, respectively (Fisher's protectedleast significant difference (FPLSD): p<0.0001); however, waveforms withanodic phase first were more efficient for PW_(cathodic)≧0.5 ms and 0.2for PW_(anodic)/PW_(cathodic)=1 and 5, respectively, and for 0.1ms≦PW_(cathodic)≦0.5 ms for PW_(anodic)/PW_(cathodic)=10 (FPLSD:p<0.0001); and energy efficiency improved as PW_(anodic)/PW_(cathodic)increased (FPLSD: p<0.0001). FIGS. 14A to 14H show that, compared to themonophasic GA waveforms, the biphasic GA waveforms were lessenergy-efficient, but the difference in energy efficiency decreased asPW_(cathodic) increased.

DESCRIPTION OF THE PREFERRED EMBODIMENTS I. System Overview

FIG. 3 is a system 10 for stimulating tissue of the central nervoussystem. The system includes a lead 12 placed in a desired position incontact with central nervous system tissue. In the illustratedembodiment, the lead 12 is implanted in a region of the brain, such asthe thalamus, subthalamus, or globus pallidus for the purpose of deepbrain stimulation. However, it should be understood, the lead 12 couldbe implanted in, on, or near the spinal cord; or in, on, or near aperipheral nerve (sensory or motor), in any subcutaneous tissue such asmuscle tissue (including cardiac tissue) or adipose tissue for thepurpose of selective stimulation to achieve a therapeutic purpose. Inaddition, the lead 12 may be utilized for transcutaneous stimulationwhere electrodes are placed, not subcutaneous, but on an outer skinsurface.

The distal end of the lead 12 carries one or more electrodes 14 to applyelectrical pulses to the targeted tissue region. The electrical pulsesare supplied by a pulse generator 16 coupled to the lead 12.

In the illustrated embodiment, the pulse generator 16 is implanted in asuitable location remote from the lead 12, e.g., in the shoulder region.It should be appreciated, however, that the pulse generator 16 could beplaced in other regions of the body or externally to the body.

When implanted, at least a portion of the case or housing of the pulsegenerator can serve as a reference or return electrode. Alternatively,the lead 12 can include a reference or return electrode (comprising abi-polar arrangement), or a separate reference or return electrode canbe implanted or attached elsewhere on the body (comprising a mono-polararrangement).

The pulse generator 16 includes stimulation generation circuitry, whichpreferably includes an on-board, programmable microprocessor 18, whichhas access to and/or carries embedded code. The code expressespre-programmed rules or algorithms under which desired electricalstimulation is generated, having desirable electrical stimulationparameters that may also be calculated by the microprocessor 18, anddistributed to the electrode(s) 14 on the lead 12. According to theseprogrammed rules, the pulse generator 16 directs the stimulation throughthe lead 12 to the electrode(s) 14, which serve to selectively stimulatethe targeted tissue region. The code may be programmed, altered orselected by a clinician to achieve the particular physiologic responsedesired. Additionally or alternatively to the microprocessor 18,stimulation generation circuitry may include discrete electricalcomponents operative to generate electrical stimulation having desirablestimulation parameters. As shown in FIG. 2, the stimulation parametersmay include a pulse amplitude (expressed, e.g., in a range from 10 μAupwards to 10 mA); a pulse width (PW) or duration (expressed, e.g., in arange from 20 μs upwards to 500 μs); a frequency of stimulation pulsesapplied over time (expressed, e.g., in a range from 10 Hz upwards to 200Hz); and a shape or waveform of the stimulation pulses. One or more ofthe parameters may be prescribed or predetermined as associated with aparticular treatment regime or indication.

In the illustrated embodiment, an on-board battery 20 supplies power tothe microprocessor 18 and related circuitry. Currently, batteries 20must be replaced every 1 to 9 years, depending on the stimulationparameters needed to treat a disorder. When the battery life ends, thereplacement of batteries requires another invasive surgical procedure togain access to the implanted pulse generator. As will be described, thesystem 10 makes possible, among its several benefits, an increase inbattery life.

As will be described in greater detail later, the stimulationparameters, which may be prescribed, used by the pulse generator differfrom conventional stimulation parameters, which may be prescribed, inthat the waveform shape of the pulses has been optimized by use of anoptimization algorithm, such as a global optimization algorithm. Anexample of a global optimization algorithm used to optimize anelectrical stimulation waveform is a genetic algorithm (GA) used tooptimize energy efficiency of a waveform for neural stimulation. Use ofthe waveform shapes optimized for energy-efficiency leads to a decreasein power consumption, thereby prolonging battery life, reducing batterysize requirements, and/or reducing frequency of battery replenishment.

Although the following description is based largely on a geneticalgorithm, other optimization algorithms may be employed in acomputational model of neural stimulation to optimize the stimulationbased on a cost function, which can include a variety of factors, suchas energy efficiency. Other optimization algorithms that may be usedinclude, for example, simulated annealing, Monte-Carlo methods, otherevolutionary algorithms, swarm algorithms (e.g. ant colony optimization,bees optimization, particle swarm), differential evolution, fireflyalgorithm, invasive weed optimization, harmony search algorithm, and/orintelligent water drops.

II. Energy-Optimal Waveforms (Monophasic)

A. Overview

The inventors have implemented a genetic algorithm in a computationalmodel of peripheral nerve stimulation, to determine the energy-optimalwaveform shape for neural stimulation. The energy efficiencies of the GAwaveforms were compared to those of conventional waveform shapes in acomputational model of a population of axons as well as during in vivostimulation of peripheral nerve fibers.

B. Deriving the Genetic Algorithms

1. Generally

The genetic algorithm seeks an optimal solution through a process basedon the principles of biological evolution. As shown in FIG. 4A, thefirst generation of a GA starts with a population of candidatesolutions. In FIG. 4A, there are two candidate stimulation parameters,each having a different wave form (rising ramp and square). Thecandidate solutions are analogous to natural organisms, and thestimulation parameters that characterize each candidate are its “genes”.

Next, as further shown in FIG. 4A, the fitness of each solution isassessed using a cost function specific to the optimization problem. Aswill be described in greater detail later, the fitness is assessed in acomputational model of extracellular stimulation of a single myelinatedmammalian peripheral axon. The fitness of each candidate (n) isexpressed in terms of energy efficiency (Energy_(n)).

As shown in FIG. 4B, the candidate solutions “mate” with each other,resulting in offspring solutions that possess a combination of theparents' genes (i.e., stimulation parameters), as well as, in time, thegenes of the offspring that have mutated (a different stimulationparameter value, preferably not found in the parents). The fitness ofboth the mating process and mutations promote a thorough search of thesolution space, to improve the chances of discovering the global optimumrather than a local optimum. Following each generation, the populationis partially or completely replaced by the offspring. As the GAprogresses, beneficial genes remain in the gene pool of the populationwhile unfavorable genes are discarded.

As shown in FIG. 4C, this process—evaluating fitness, mating, andreplacing solutions—is repeated either for a predetermined number ofgenerations (such as 10, 20, 50, 100, 200, 500, 1000, 2000, 5000,10,000, or more generations) or until the solutions converge upon,towards, or within a desirable range from a fitness value. The solutionwith the overall greatest fitness is the resulting estimate of theoptimal solution.

2. The Specific Genetic Algorithm

A specific generic algorithm (GA) was derived to seek the energy-optimalwaveform shape in a computational model of extracellular stimulation ofa single myelinated mammalian peripheral axon, which is shown in FIGS.5A and 5B.

Simulations were run in NEURON (Hines and Carnevale 1997) using the MRGmodel (fiber diameter=11.5 μm), which represented a myelinated mammalianperipheral axon as a double cable model with a finite impedance myelinsheath and explicit representation of the nodes of Ranvier, paranodalsections, and internodal segments (McIntyre et al. 2002) (see FIG. 5B).Stimulation was delivered through a current-regulated point sourcesituated within a conductive medium (300 Ω-cm) (McNeal 1976) located 1mm directly above the center node of the fiber (see FIG. 5A).

C. Deriving the GA Waveforms

FIG. 6A shows an overview of the results of the GA waveform derivationprocess.

For each generation of the GA, the population consisted of fifty (50)stimulation waveforms with fixed pulse width (PW). Waveforms werediscretized in time using a time step equal to that of the computationalmodel (dt=0.002 ms), and the genes of each waveform represented theamplitudes at every time step. The values of the genes of the waveformsof the first generation were selected at random from a uniformdistribution between zero and two times the cathodic threshold ofstimulation with a rectangular waveform at the equivalent PW (e.g., 807μA for PW=10 μs; 190 μA for PW=100 μs; 79.8 μA for PW=1 ms).

The cost function (F) used to evaluate the fitness of each waveformequaled the sum of the energy consumed by the waveform (E) and asubstantial penalty if the waveform failed to elicit an actionpotential:

F=E+Penalty  Equation (1)

$\begin{matrix}{E = {{\int_{0}^{PW}{{P(t)}\ {t}}} \propto {{t}*{\sum\limits_{n = 1}^{N}I_{n}^{2}}}}} & {{Equation}\mspace{14mu} (2)}\end{matrix}$

where P is instantaneous power, t is time, I is the instantaneouscurrent, and N is the number of discretizations (genes) of a stimulationwaveform. If the waveform elicited an action potential, then Penaltyequaled 0, but if the waveform did not elicit an action potential, thenPenalty equaled 1 nJ/ohm (2 to 3 orders of magnitude larger than E).

At the end of each generation, the top ten (10) fittest waveforms (i.e.,smallest F) remained in the population while the remaining forty (40)waveforms were replaced by offspring. Every waveform, regardless of itsvalue of F, had an equal probability of being selected as a parent, andeach offspring was generated by combining the genes of two parents usingtwo crossover points. A crossover point was a randomly selected genelocation, where during mating the genes prior to the crossover pointfrom one parent were combined with the genes beyond the crossover pointfrom the other parent. With two crossover points, the effect was a swapof a segment of one parent's genes with the corresponding section of theother parent's genes.

Each gene of the offspring was mutated by scaling the value by a randomfactor chosen from a normal distribution (μ=1, σ²=0.025). Because theinitial waveforms were monophasic cathodic pulses, the genes wererestricted to negative values.

The GA was run using a wide range of PWs (0.02, 0.05, 0.1, 0.2, 0.5, 1,and 2 ms) to determine whether the outcome of the GA varied with PW. Foreach PW, the GA was run for 5 independent trials of 10,000 generationswith different initial populations. For each trial, the following wasrecorded: the energy consumed by the most energy-efficient waveform ofeach generation (generation energy); the most energy-efficient waveformof the final generation (GA waveform); and the charge (Q) delivered bythe GA waveform, where:

$\begin{matrix}{Q = {{\int_{0}^{PW}{{I(t)}\ {t}}} = {{t}*{\sum\limits_{n = 1}^{N}{I_{n}.}}}}} & {{Equation}\mspace{14mu} (3)}\end{matrix}$

For each PW, the means and standard errors of the energy and chargeconsumed by the GA waveforms across the 5 independent trials wererecorded.

In this particular GA, the only considerations for the cost function (F)were energy efficiency and whether or not an action potential waselicited in the axon. However, F can use other measures besides energyefficiency, either as the sole consideration of F, or in combinationwith other measures. These other measures may include charge efficiency,power efficiency (i.e., peak power of waveform), maximum voltage orcurrent, therapeutic benefit of stimulation, adverse effects, andselectivity of stimulation (i.e., activation of one population ofneurons or fibers—defined by location, size, or type—without activationof other populations). F may include different weights associated witheach measure, reflecting the relative importance of each measure. Forexample, F may consider both energy and charge, and energy may be threetimes as important as charge for a given application of stimulation.Then, F=0.75 E+0.25 Q. Thus, the methodology according to the presentinvention may produce waveform shapes that may be optimized to anyparticular cost function, including desired cost parameters.

D. The Resulting GA Waveforms

Each trial of the GA began with a different population of randomwaveforms, but by the end of each trial, the GA converged uponconsistent and highly energy-efficient waveform shapes (as FIGS. 6A and6B show). The generation energy converged to within 1% of the finalgeneration energy by 5000 generations for PW≦0.5 ms and by 9000generations for PW=1 and 2 ms. As FIG. 7 shows, for each PW, the GAwaveforms were very similar across trials, and across PWs the shapes ofthe GA waveforms were quite similar. As FIG. 6 shows, for PW≦0.2 ms, theGA waveforms resembled truncated Gaussian curves, with the peak near themiddle of the pulse. For PW≧0.5 ms, the shapes of the GA waveforms alsoresembled Gaussian curves but with leading and/or trailing tails ofnegligible amplitude.

E. Assessing the Energy-Efficiency of GA Waveforms

1. The Population Model

(i) Methodology

The GA waveforms were evaluated in a population model of one hundred(100) parallel MRG axons (11.5-μm diameter) distributed uniformly withina cylinder with a 3-mm diameter. Extracellular stimulation was deliveredthrough a point current source located at the center of the cylinder.For each PW (0.02, 0.05, 0.1, 0.2, 0.5, 1, and 2 ms), ten (10)populations of randomly-positioned axons were selected. For eachpopulation, input/output (I/O) curves (see FIG. 8) were constructed ofthe number of fibers activated vs. E, as well as the number of fibersactivated vs. Q. To adjust the stimulation amplitude of a waveform, theentire waveform was scaled. For each I/O curve, the E and Q needed toactivate 50% of the entire population were computed, and the means andstandard errors of these values across the ten (10) axon populationswere calculated. Using the same axon populations, I/O curves forconventional waveforms used in neural stimulation were calculated:rectangular, rising/decreasing ramp, rising/decaying exponential, andsinusoid waveforms (See Appendix for the equations for the conventionalwaveforms).

(ii) Results

(a) Overview

The GA waveforms were more energy-efficient than the conventionalstimulation waveform shapes for all PWs in the population models. Theenergy-duration curve of the GA waveforms was concave up (see FIG. 9A),and the minimum E for the GA waveforms across PWs was less than theminimum E for the conventional waveform shapes. Of these other shapes,the shape that most resembled the GA waveforms—the sinusoid—had thelowest minimum energy across PWs. For PW≦0.2 ms, the GA waveforms wereslightly more energy-efficient (<20%) than the other waveform shapes(see FIG. 9B). Between PW=0.2 ms and 0.5 ms, the differences in energyefficiency between GA waveforms and the conventional shapes increasedconsiderably, and these differences increased further with PW for allbut the exponential waveforms. Because the positions of the axons wererandomized in the population model, these results demonstrate that thesuperior energy efficiency of the GA waveforms was independent of thelocation of the electrode with respect to the axon.

The GA waveforms were also more energy-efficient than most of thewaveform shapes when energy was plotted against charge. For all waveformshapes, the curves of E vs. Q were concave up and many of the curvesoverlapped substantially (see FIG. 9C). However, the curves for the GAwaveforms and sinusoid lay under the other curves, indicating that for agiven amount of charge, the GA and sinusoid waveforms consumed lessenergy to reach threshold than the other waveform shapes.

(b) GA Waveform Sensitivity Analysis

As shown in FIG. 10A, the energy-optimal waveform shapes were largelyinsensitive to variations in the parameters of the GA. Doubling orhalving the number of waveforms that survived to the next generation orthe number of waveforms in each generation had no substantial effects onthe shape of the GA waveforms or their energy efficiencies (<0.1%difference). Also, the amplitudes of the waveforms in the initialgeneration were scaled between 0.4-1.6 times the original amplitudes,and scaling factors>0.8 had little effect on the shape and energyefficiency (<0.1% difference) of the GA waveform. Scaling factors below0.6, however, resulted in initial waveforms that were all belowthreshold, and the GA did not converge to an energy-efficient waveform.In addition, the variance of the normal distribution used in mutationswas scaled between 0-4 times the original variance. With variance=0 (nomutations), the GA rapidly converged on an energy-inefficient waveform.However, for all other values of variance the GA produced nearlyidentical GA waveforms with approximately the same energy efficiencies(<0.4% difference).

As shown in FIG. 10B, although the shape of the GA waveforms remainedconsistent when dt was varied between 0.001-0.01 ms, the energyefficiency did change. Smaller values of dt produced finer resolution ofthe waveform shape, which created more energy-efficient GA waveforms forPW≦0.1 ms (|ΔE|<110). However, the improved resolution also led to lessenergy-efficient GA waveforms for PW≧1 ms, as a result of more noise inthe waveform (|ΔE|<10.5%).

In addition to using a fiber diameter of 11.5 μm, we ran the GA withfiber diameters of 5.7 μm and 16 μm. The GA waveforms produced for eachfiber diameter remained the most energy-efficient waveforms in theirrespective models, and their overall shapes were consistent acrossdiameters (see FIG. 10A). Further, the GA waveforms optimized fordiameter=11.5 μm (see FIG. 7) were still more energy-efficient than theconventional waveform shapes for excitation of the other two diameters.

The shape and efficiency of GA waveforms were dependent on the model ofthe neural membrane. We ran the GA in a model of a myelinated axon thatconsisted of nodes with Hodgkin-Huxley membrane parameters connected byelectrically insulated myelinated internodes. This model differed fromthe MRG model both geometrically (e.g., no paranodal sections) andphysiologically (e.g., lower temperature, no persistent sodiumchannels), but the fiber diameter and electrode-fiber distance wereunchanged. In the Hodgkin-Huxley model, for PW≦0.05 ms the GA waveformsgenerated in the Hodgkin-Huxley model were still unimodal as in the MRGmodel but were asymmetric (see FIG. 10B). However, for PW=0.02 ms the GAwaveforms from the two models diverged (see FIG. 10C). In addition, whentested in the Hodgkin-Huxley model, the original GA waveforms from theMRG model were not uniformly more energy-efficient than conventionalwaveform shapes.

(c) GA Waveform Fit with Analytical Equation

To gain a better understanding of the exact shapes of theenergy-optimized waveforms, the GA waveforms were fitted to a piece-wisegeneralized normal distribution:

$\begin{matrix}\begin{matrix}{{f(t)} = {{A*^{- {(\frac{{t - \mu}}{\alpha_{L}})}^{\beta_{L}}}{for}\mspace{14mu} t} \leq \mu}} \\{= {{A*^{- {(\frac{{t - \mu}}{\alpha_{R}})}^{\beta_{R}}}\mspace{14mu} {for}\mspace{14mu} t} > {\mu.}}}\end{matrix} & {{Equation}\mspace{14mu} (4)}\end{matrix}$

where A is the amplitude at the peak, located at t=μ; α's and β's arescale and shape parameters, respectively, and must be greater than zeroand the α's and β's are preferably less than infinity; and thesubscripts correspond to the left (L) and to the right (R) of the peak.When α_(L)=α_(R) and β_(L)=β_(R), the function is symmetric about β, andthe values of β dictate the kurtosis (i.e., peakedness) of the waveform.When α_(L)≠α_(R) and/or β_(L)≠β_(R), varying degrees of kurtosis andskewness can be produced [see Appendix for the equations]. Thus,Equation (4) may be used to produce an energy-optimal electricalstimulation waveform.

The parameters of Equation (4) were optimized to fit the mean GAwaveforms (i.e., as shown FIG. 7) using the lsqcurvefit function inMatlab (R2007b; The Mathworks, Natick, Mass.). The least-squareoptimized waveforms fit well with the energy-optimized waveforms(R²>0.96). Across PWs, the fitted waveforms were not very skewed(−0.5<skewness<0.5, where skewness=0 is perfect symmetry), had sharperpeaks (kurtosis>0.55) than the normal distribution (kurtosis=0), and thekurtosis of the fitted waveforms increased with PW.

A modified GA was also run, where the stimulation waveforms werecharacterized by Equation (4) instead of by the amplitudes at each timestep. As a result, all waveforms were characterized by only sixparameters—A, μ, α_(L), α_(R), β_(L), and β_(R)—and initial values ofthese parameters were selected at random from uniform distributions (A:between zero and four times the cathodic threshold of stimulation with arectangular waveform at equivalent PW; μ: 0−PW; α's: 0.01-0.5; β's:0.01-3).

Preferably, a waveform defined at least in part by Equation (4) isgenerated or controlled by a microprocessor, which may accept variousvalues for the indicated parameters. The peak current amplitude (A)varies with the stimulation application, and may vary between patients,but as described earlier, typically ranges from about 10 μA to about 10mA. Parameter μ is preferably between zero (0) and the stimulation pulsewidth (PW). Parameters α_(L), α_(R), β_(L), and β_(R) are preferablygreater than 0 and less than infinity. One exemplary set of preferredalpha and beta values for a monophasic GA waveform is alpha values inthe range of about 0.008 milliseconds to about 0.1 milliseconds and betavalues in the range of about 0.8 to about 1.8. However, the alpha andbeta values may well fall outside of this range under differentcircumstances, and changes in the values may be directly associated witha given fiber diameter.

The GA waveforms that resulted from optimization with this modified GAwere not substantially different from the waveforms generated by theinitial GA. The shapes of the waveforms were quite similar to theinitial GA waveforms across all PWs (R²>0.93), and the energyefficiencies improved very little (<2%) for PW≦0.5 ms. However, themodified GA waveforms were more energy-efficient than the initial GAwaveforms for PW=1 and 2 ms (5.6% and 10.4%, respectively), as a resultof the smoothness of the modified GA waveforms and their ability toreach amplitudes near zero at the tails. Consequently, theenergy-duration curve with this GA was not concave up as with theoriginal GA, but instead, E remained constant as PW increased.

2. In Vivo Experiments

(i) Surgical Preparation

All animal care and experimental procedures were approved by theInstitutional Animal Care and Use Committees of Duke University and werefollowed according to The Guide to the Care and Use of LaboratoryAnimals, 1996 Edition, National Research Council.

Experiments were performed on 3 adult male cats. Sedation was inducedwith acepromazine (Vedco Inc., 0.3 mg/kg; S.Q.), and anesthesia wasinduced with ketamine HCl (Ketaset 35 mg/kg; I.M.) and maintained duringthe experiment with α-chloralose (Sigma-Aldrich, Inc., initial 65 mg/kgsupplemented at 15 mg/kg; I.V.). The cat was intubated, and respirationwas controlled to maintain end tidal CO2 at 3-4%. Core temperature wasmonitored and maintained at 39° C. Fluid levels were maintained withsaline solution and lactated ringers delivered through the cephalic vein(15 ml/kg/hr, I.V.). Blood pressure was monitored using a catheterinserted into the carotid artery.

The sciatic nerve was accessed via an incision on the medial surface ofthe upper hindlimb. As FIG. 11A shows, a monopolar cuff electrode,composed of a platinum contact embedded in a silicone substrate, wasplaced around the nerve and secured with a suture around the outside ofthe electrode. The return electrode was a subcutaneous needle. Twostainless steel wire electrodes were inserted into the medialgastrocnemius muscle to measure the electromyogram (EMG) evoked bystimulation of the sciatic nerve (see FIG. 11B). The EMG signal wasamplified, filtered (1-3000 Hz), recorded at 500 kHz, rectified, andintegrated to quantify the response (EMG integral).

Stimulation and recording were controlled with Labview (DAQ:PCI-MIO-16E-1) (National Instruments, Austin, Tex.). A voltage waveformwas delivered at a rate of 500 ksamples/s to a linear voltage-to-currentconverter (bp isolator, FHC, Bowdoin, ME) and delivered through the cuffelectrode. The voltage across (V) and current through (I) the cuffelectrode and return electrode were amplified (SR560, Stanford ResearchSystems, Sunnyvale, Calif.) and recorded (fsample=500 kHz). The energydelivered during stimulation was determined by integrating the productof V(t) and I(t):

E=∫ ₀ ^(PW) P(t)dt=∫ ₀ ^(PW) V(t)I(t)dt  Equation (5)

The charge delivered during stimulation was determined by integratingI(t) using Equation (3), above.

(ii) Recruitment Curves

Recruitment curves of the integral of the rectified EMG as a function ofE and Q were measured for rectangular, decaying exponential (timeconstant [τ]=132, 263, and 526 μs), and GA waveforms at various PWs(0.02, 0.05, 0.1, 0.2, 0.5, and 1 ms) in random order. At frequentintervals over the course of the experiment, stimulation with therectangular waveform at a fixed PW was provided to monitor shifts inthreshold. Threshold shifts occurred in only one animal, and the valuesof E and Q were scaled accordingly. Recruitment curves were generatedusing a similar procedure as in the computational models: stimulusamplitude was incremented, three (3) stimulation pulses were deliveredat −1 Hz at each increment, and the average values of E, Q, and EMGintegral were recorded. From each recruitment curve, the values of E andQ required to generate 50% of the maximal EMG were calculated, and thevalues at PW=0.02 ms for the rectangular waveform were defined asbaseline values. Subsequently, all values of E and Q were normalized totheir respective baseline value, and the means and standard errorsacross experiments were calculated.

After log-transformation of the data, the effects of waveform shape onenergy and charge efficiency were analyzed. A two-way repeated measuresANOVA was performed for each measure of efficiency; the dependentvariable was E or Q, and the independent variables were waveform shape,PW (within-subjects factors), and cat (subject). Where interactionsbetween waveform shape and PW were found to be significant (p<0.05), thedata were subdivided by PW for one-way repeated measures ANOVA. Again,the dependent variable was E or Q, and the independent variables werewaveform shape (within-subjects factor) and cat (subject). For testswhich revealed significant differences among waveforms (p<0.05), posthoc comparisons were conducted using Fisher's protected leastsignificant difference (FPLSD). Although data were log-transformed forstatistical analysis, data were plotted as average percent differencewith respect to the GA waveforms.

(iii) Results

The in vivo measurements comparing the efficiency of GA waveforms torectangular and decaying exponential waveforms largely corroborated theresults of the population model. For PW 0.05 ms, the GA waveforms weresignificantly more energy-efficient than most of the rectangular anddecaying exponential waveforms (p<0.05, FPLSD) (see FIGS. 12A and 12B).Although the decaying exponential with τ=132 μs appeared to be moreenergy-efficient than the GA waveforms for PW≧0.5 ms, this result wasmisleading; for long PWs, increasing the PW for exponential waveformssimply extends the low-amplitude tail, which has negligible effects onexcitation. As a result, the energy-duration curve for the exponentialwaveforms leveled off at long PWs, while the energy-duration curve forthe GA waveforms increased with PW, as in the population model. Whennormalized E was plotted against normalized Q, the GA waveforms appearedto be more energy-efficient than the rectangular waveform for normalizedQ>2 (see FIG. 12C). However, the GA waveforms were not substantiallymore energy-efficient than the decaying exponential waveforms.

III. Energy-Optimal Waveforms (Biphasic)

The original GA revealed energy-optimal waveforms for monophasicstimulation. However, most waveforms used for nerve stimulation arebiphasic. Because the charge recovery pulse can influence the thresholdof the primary pulse (van den Honert and Mortimer 1979), it washeretofore unclear whether the monophasic GA waveforms would remainenergy-optimal for biphasic stimulation. First, thresholds wererecalculated in the single fiber model for all waveform shapes with theaddition of rectangular charge-balancing anodic phases. The duration wasvaried (PW_(anodic)/PW_(cathodic)=1, 5, or 10), as was the timing(preceding or following the cathodic phase) of the charge-balancingphase. Amplitudes of the anodic phases were adjusted to produce zero netcharge for the entire waveform, and E was calculated from both phases ofthe waveform.

The biphasic results showed that the GA waveforms optimized formonophasic stimulation were not the most energy-efficient waveformsacross all PWs. Therefore, the GA was modified to seek energy-optimalbiphasic waveform shapes. For each combination of duration and timing(i.e., before or after the cathodic phase) of the rectangularcharge-balancing anodic phase, five (5) separate trials were run of theGA to optimize the shape of the cathodic pulse for PW=0.02−1 ms, and Ewas calculated from both the anodic and cathodic phases of the waveform.

The shapes of biphasic GA waveforms varied with both the timing andduration of the anodic phase. Most waveforms still resembled truncatednormal curves, but the peaks of the cathodic phases were shifted awayfrom the anodic phase (see FIG. 13). As with the monophasic GAwaveforms, as PW_(cathodic) increased the waveforms generally becameflatter. The duration of the anodic phase relative to the cathodic phaseinfluenced the peakedness of the resulting waveforms: the shorter theanodic phase, the sharper the peak of the cathodic phase. However, forwaveforms with anodic phase first, PW_(anodic)=1 ms andPW_(cathodic)=0.2 or 0.1 ms, the peaks of the resulting waveforms weresharper than expected. Surprisingly, the peaks of both of thesewaveforms were located exactly 0.086 ms after the anodic pulse for everytrial. Analysis of the gating parameters and membrane voltage duringstimulation did not reveal any apparent explanations for this particularshape.

The biphasic GA waveforms were applied to five randomly selectedpopulations from the population model, and energy-duration curves werecalculated as in the monophasic case. Energy efficiencies of thebiphasic GA waveforms as well as conventional waveforms were dependenton the timing and duration of the anodic phase (see FIGS. 14A and 14B).Conventional waveform shapes were paired with rectangularcharge-balancing anodic phases with the same duration and timing as thebiphasic GA waveforms, and the energy efficiencies of these waveformswere calculated in the population model. The biphasic GA waveforms werealways more energy-efficient than the conventional waveform shapes, andthe differences in energy efficiency varied with the duration of theanodic phase. In general, as PW_(anodic)/PW_(cathodic) increased thedifference in energy efficiency between the biphasic GA waveforms andthe conventional waveform shapes decreased (see FIGS. 14C to 14H). Aswell, for PW_(anodic)/PW_(cathodic)=1 the differences between thebiphasic GA waveforms and the conventional waveforms were generallygreater than the differences in the monophasic case (FIGS. 8A to 8C),but for PW_(anodic)/PW_(cathodic)=10 the differences were smaller thanin the monophasic case.

IV. Conclusion

The genetic algorithm (GA) described herein mimics biological evolution,to provide an optimal energy-efficient waveform shape for neuralstimulation. The GA generates highly energy-efficient GA waveforms thatresembled truncated Gaussian curves. When tested in computationalmodels, and as confirmed by in vivo peripheral nerve stimulation, the GAwaveforms are more energy-efficient than many conventional waveformshapes. The differences in energy-efficiency are more substantial forlong PWs than for short PWs. The GA waveforms will extend the batterylife of implantable stimulators, and thereby reduce the costs and risksassociated with battery replacements, decrease the frequency ofrecharging, and reduce the volume of implanted stimulators.

Along with energy efficiency, the charge efficiency of stimulation is animportant consideration with implanted devices. The charge deliveredduring a stimulus pulse contributes to the risk of tissue damage (Yuenet al. 1981; McCreery et al. 1990). Charge efficiency can beincorporated into the cost function, F (Equation (1)), with weightsassociated with charge and energy efficiency that reflected the relativeimportance of each factor. Charge efficiency was not considered in F inthe GA described herein. Nevertheless, the GA waveforms ended up beingsimultaneously energy- and charge-efficient.

In the computational models, the GA waveforms were the mostenergy-efficient waveform shapes. All five independent trials of the GAconverged to nearly the same shape for each PW and achieved similarlevels of energy efficiency. In addition, all GA waveforms resembletruncated Gaussian curves, and none of the variations in the parametersof the GA had substantial effects on the outcome.

The energy efficiencies of non-GA Gaussian or sinusoids have beeninvestigated previously. Sahin and Tie (2007) found in a computationalmodel of a mammalian myelinated axon (Sweeney et al. 1987) that Gaussianand sinusoid waveforms had the lowest threshold energies out of severalconventional waveform shapes. However, unlike the GA waveforms describedherein, the Gaussian and sinusoid waveforms were not the mostenergy-efficient waveforms across all PWs. Qu et al. (2005) conducted invitro experiments on rabbit hearts and found that defibrillation wasachieved with significantly less energy for Gurvich (biphasic sinusoid)waveforms than with biphasic decaying exponential or rectangularwaveforms. Dimitrova and Dimitrov (1992) found in a model of anunmyelinated Hodgkin-Huxley axon that waveforms that resembledpostsynaptic potentials (skewed Gaussian) were more energy-efficientthan rectangular waveforms. Although these previous studies showed thatthe sinusoid, Gaussian, or skewed Gaussian waveforms were moreenergy-efficient than other waveform shapes, these non-GA waveforms werenot proven to be energy-optimal.

The GA with genes representing the parameters of the piece-wisegeneralized normal distribution (Equation (4)) did not produce GAwaveforms with noticeably different shapes. However, the waveforms weremuch smoother, and for long PWs the tails were much closer to zero.These differences improved the energy efficiency over the original GAwaveforms, particularly for long PWs. As a result, the energy-durationcurve was no longer concave up, as in the original GA (see FIG. 8A), butinstead E never increased as PW increased. This result is moreconsistent with expectations; one would expect that at a given PW the GAcould produce any waveform that was produced at a shorter PW bounded bytails of zero amplitude. Therefore, as PW increases E should eitherlevel off or decrease.

The different properties of the MRG axon and the Hodgkin-Huxley axon ledto the dissimilarities in the genetically-optimized waveforms producedin the two models. Not only were the differences in ion channel dynamicsbetween the two models substantial, but also the Hodgkin-Huxley axonlacked paranodal sections, and both factors likely contributed to thedifferences in GA waveforms. However, due to the non-linearity andcomplexity of the equations governing membrane voltage, it is difficultto pinpoint which characteristics of the axonal models were mostresponsible for the varying results. Additional trials of the GA inmodels where specific geometric and physiological parameters were variedsystematically could determine how energy-optimal waveforms change withmodel parameters. Thus, the GA approach can determine energy-optimalwaveform shapes for a given model or system, but the optimal shape maybe different in each case.

The biphasic GA waveforms exhibited many similarities to the monophasicGA waveforms. Both sets of GA waveforms were more energy-efficient thanseveral conventional waveform shapes and were unimodal in shape.However, the peakedness and locations of the peaks of the biphasic GAwaveforms were different than the monophasic GA waveforms. The effectsof the anodic phase on the sodium channels explain many of thedifferences among the shapes of the biphasic GA waveforms. The anodicphase hyperpolarizes the membrane, deactivating m-gates andde-inactivating the h-gates of the sodium channel. When the cathodicphase was delivered first, the peak likely shifted away from the anodicphase to activate the sodium channels earlier than in the monophasiccase, thus offsetting the deactivation generated by the anodic phase.When the anodic phase was delivered first, the peak shifted away fromthe anodic phase to allow the m-gate of the sodium channels to return tobaseline. Differences between the monophasic and biphasic GA waveformswere greater for short PW_(anodic) than for long PW_(anodic). AsPW_(anodic) increased, the amplitude of the anodic phase decreased,reducing the effect of the anodic phase on membrane voltage and thesodium channels. Consequently, the biphasic GA waveforms began toresemble the monophasic GA waveforms in both shape and energyefficiency.

The foregoing description describes the energy and charge efficiency forexcitation of peripheral nerve fibers. Still, the technical features ofthe GA waveforms pertain to stimulation of other components of thenervous system. During spinal cord stimulation, the targets ofstimulation are thought to be axons (Coburn 1985; Struijk et al. 1993;Struijk et al. 1993), and the current findings would likely beapplicable. As well, our results would be valid for muscularstimulation, where the targets of stimulation are motor nerve axons(Crago et al. 1974). The technical features for the GA waveforms asdescribed herein can also be relevant for stimulation of the brainbecause in both cortical stimulation (Nowak and Bullier 1998; Manola etal. 2007) and deep brain stimulation (McIntyre and Grill 1999), thetargets of stimulation are thought to be axons.

GA waveforms could substantially increase the battery life of implantedstimulators. For example, the stimulators used for deep brainstimulation last approximately 36-48 months with conventional waveforms(Ondo et al. 2007). Over 30 years, the device would have to be replacedabout 8-10 times. Over a clinically relevant range of PWs (˜0.05-0.2 ms)the GA waveforms were upwards to approximately 60% more energy-efficientthan either the rectangular or decaying exponential waveforms, which arethe most frequently used waveforms clinically (Butson and McIntyre2007). A 60% improvement in energy efficiency would extend battery lifeby over 21 months. As a result, over 30 years the device would only haveto be replaced about 5-6 times.

The GA described herein did not account for the energy consumed by theelectronic circuitry of an implantable stimulator. A stimulationwaveform that can be generated using a simple analog circuit may consumeless energy than a waveform that requires several active components. Ifthe energy consumption of the circuitry were incorporated into the GA,then the algorithm may produce different waveform shapes.

Various features of the invention are set forth in the claims thatfollow.

APPENDIX 1. Conventional Waveform Shapes

Thresholds were measured for conventional waveforms used in neuralstimulation: rectangular, rising/decreasing ramp, rising/decayingexponential, and sine wave. For all shapes, stimulation was applied att=0 and turned off at t=PW. The equation for the stimulus current withthe rectangular waveform was

I _(stim)(t)=K _(s) *[u(t)−u(t−PW)]  Equation (6)

where K_(s) is the current amplitude, t is time, and u(t) is the unitstep function. The equations for the rising and decreasing ramp were

I _(stim)(t)=K _(r) *t*[u(t)−u(t−PW)]  Equation (7)

I _(stim)(t)=K _(r)(PW−t)*[u(t)−u(t−PW)]  Equation (8)

respectively, where K_(r) is the magnitude of the slope of the ramp. Theequations for the rising and decaying exponential waveforms were

I _(stim)(t)=K _(e) e ^(t/τ) *[u(t)−u(t−PW)]  Equation (9)

I _(stim)(t)=K _(e) e ^((PW-t)/τ) *[u(t)−u(t−PW)]  Equation (10)

respectively, where K_(e) is the amplitude at t=0 for Equation (9) andat t=PW for Equation (10). In the computational models, τ equaled 263μs. The equation for the sine wave was

$\begin{matrix}{{I_{stim}(t)} = {K_{\sin}*{\sin \left( {\frac{t}{PW}\pi} \right)}*\left\lbrack {{u(t)} - {u\left( {t - {PW}} \right)}} \right\rbrack}} & {{Equation}\mspace{14mu} (11)}\end{matrix}$

where K_(sin) is the amplitude of the sine wave. Note that only one halfof one period of the sine wave is delivered during the pulse.

2. Skewness and Kurtosis of Piece-Wise Generalized Normal Distribution

To quantify the shape of the GA waveforms, the waveforms were fitted toa piece-wise generalized normal distribution, f(t) (4), and calculatedthe skewness and kurtosis. First, the peak was centered about t=0:

τ=t−μ  Equation (12).

Then, f(τ) was normalized so the time integral from −∞ to +∞ equaled 1:

$\begin{matrix}\begin{matrix}{N = {\int_{- \infty}^{\infty}{{f(\tau \ )}{\tau}}}} \\{= {{\int_{- \infty}^{0}{{f_{L}(\tau)}\ {\tau}}} + {\int_{0}^{\infty}{{f_{R}(\tau)}\ {\tau}}}}} \\{= {{\alpha_{L}{\Gamma \left( {1 + \frac{1}{\beta_{L}}} \right)}} + {\alpha_{R}{\Gamma \left( {1 + \frac{1}{\beta_{R}}} \right)}}}}\end{matrix} & {{Equation}\mspace{14mu} (13)}\end{matrix}$F(τ)=f(τ)/N  Equation (14).

Next, the mean and variance of the distribution were calculated:

$\begin{matrix}\begin{matrix}{\overset{\_}{\tau} = {\int_{- \infty}^{\infty}{\tau*{F(\tau)}\ {\tau}}}} \\{= \frac{{{- \alpha_{L}^{2}}{\Gamma \left( {1 + \frac{2}{\beta_{L}}} \right)}} + {\alpha_{R}^{2}{\Gamma \left( {1 + \frac{2}{\beta_{R}}} \right)}}}{2\; N}}\end{matrix} & {{Equation}\mspace{14mu} (15)} \\\begin{matrix}{{\sigma 2} = {\int_{- \infty}^{\infty}{\left( {\tau - \overset{\_}{\tau}} \right)^{2}*{F(\tau)}\ {\tau}}}} \\{= {\frac{\begin{matrix}{{3\; \alpha_{L}{\overset{\_}{\tau}}^{2}{\Gamma \left( {1 + \frac{1}{\beta_{L}}} \right)}} + {3\alpha_{L}^{2}\overset{\_}{\tau}\; {\Gamma \left( {1 + \frac{2}{\beta_{L}}} \right)}} +} \\{{\alpha_{L}^{3}{\Gamma \left( {1 + \frac{3}{\beta_{L}}} \right)}} + {3\; \alpha_{R}{\overset{\_}{\tau}}^{2}{\Gamma \left( {1 + \frac{1}{\beta_{R}}} \right)}} -} \\{{3\; \alpha_{R}^{2}\overset{\_}{\tau}\; {\Gamma \left( {1 + \frac{2}{\beta_{R}}} \right)}} + {\alpha_{R}^{3}{\Gamma \left( {1 + \frac{3}{\beta_{R}}} \right)}}}\end{matrix}}{3\; N}.}}\end{matrix} & {{Equation}\mspace{14mu} 16}\end{matrix}$

Finally, from these equations, skewness and kurtosis were calculated:

$\begin{matrix}{{skew} = \frac{\int_{- \infty}^{\infty}{\left( {\tau - \overset{\_}{\tau}} \right)^{3}*{F(\tau)}\ {\tau}}}{\left( \sigma^{2} \right)^{2/2}}} & {{Equation}\mspace{14mu} (17)} \\{{kurt} = {\frac{\int_{- \infty}^{\infty}{\left( {\tau - \overset{\_}{\tau}} \right)^{4}*{F(\tau)}\ {\tau}}}{\left( \sigma^{2} \right)^{2}}.}} & {{Equation}\mspace{14mu} (18)}\end{matrix}$

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1. A system for neurological tissue stimulation comprising a lead sizedand configured for implantation in a targeted tissue stimulation region,and a pulse generator coupled to the lead, the pulse generator includinga power source comprising a battery and a microprocessor coupled to thebattery and being operable to apply to the lead a stimulation waveformderived at least in part through the use of a global optimizationalgorithm.
 2. A system according to claim 1, wherein the stimulationwaveform is derived through the use of a prescribed genetic algorithm(GA) coupled to a computational model of extracellular stimulation of amammalian myelinated axon, wherein the waveform is optimized for energyefficiency to prolong battery life.
 3. A system according to claim 1wherein the waveform consists essentially of a Gaussian curve.
 4. Asystem according to claim 1 wherein the waveform consists essentially ofa truncated Gaussian curve.
 5. A system according to claim 1 wherein thewaveform is monophasic.
 6. A system according to claim 1 wherein thewaveform is biphasic.
 7. A system according to claim 1 wherein the pulsegenerator is implantable.
 8. A system according to claim 1, wherein themicroprocessor controls a generation of the stimulation waveform througha software implementation of a function comprising: $\begin{matrix}{{{f(t)} = {{A*^{- {(\frac{{t - \mu}}{\alpha_{L}})}^{\beta_{L}}}\mspace{14mu} {for}\mspace{14mu} t} \leq \mu}},{and}} \\{{{= {{A*^{- {(\frac{{t - \mu}}{\alpha_{R}})}^{\beta_{R}}}\mspace{11mu} {for}\mspace{14mu} t} > \mu}},}\;}\end{matrix}$ wherein the stimulation waveform has parameters includingamplitude and pulse width, wherein μ is greater than zero and less thanthe pulse width, wherein A is the peak current amplitude of the waveformat time μ, wherein α_(L) and α_(R) are scale parameters for times lessthan or equal to μ and greater than μ, respectively, and wherein β_(L)and β_(R) are shape parameters for times less than or equal to μ andgreater than μ, respectively.
 9. A system according to claim 8, whereinA, β, α_(L), α_(R), β_(L), and β_(R) are selectively programmable.
 10. Asystem according to claim 8, wherein A is within a range of about 10microamps to about 10 milliamps.
 11. A system according to claim 10,wherein α_(L) and α_(R) are greater than zero milliseconds and less thaninfinity milliseconds.
 12. A system according to claim 8, wherein α_(L)and α_(R) are within a range of about 0.008 milliseconds to about 0.1milliseconds.
 13. A system according to claim 12, wherein β_(L) andβ_(R) are greater than zero and less than infinity.
 14. A systemaccording to claim 13, wherein β_(L) and β_(R) are within a range ofabout 0.8 to about 1.8.
 15. A stimulation waveform for application totargeted neurological tissue, the waveform being selected from a set ofwaveforms consisting essentially of Gaussian curves having varying pulsewidths and derived by a prescribed genetic algorithm (GA) coupled to acomputational model of extracellular stimulation of a mammalianmyelinated axon, wherein the set of stimulus waveforms are optimized forenergy efficiency.
 16. A waveform according to claim 15 wherein thewaveform is monophasic.
 17. A waveform according to claim 15 wherein thewaveform is biphasic.
 18. A system for creating a stimulation waveformoptimized for energy efficiency comprising a population of parentstimulation waveforms, first means including a genetic algorithm (GA)for generating a population of offspring stimulation waveforms by matingthe population of parent stimulation waveforms, second means consistingessentially of a computational model of extracellular stimulation of amammalian myelinated axon, being operative for assessing the fitness ofindividual offspring stimulation waveforms generated by the first meansin terms of energy efficiency and selecting as a new population ofparent stimulation waveforms the current offspring stimulation waveformshaving the highest energy efficiency values, the selection terminatingwhen predetermined termination criteria is met.
 19. A system accordingto claim 18 wherein the termination criteria includes the fitness ofsuccessive populations of offspring stimulation waveforms convergingtoward a common energy-efficiency value.
 20. A system according claim 18wherein the stimulation waveform optimized for energy efficiencyconsists essentially of a Gaussian curve.
 21. A system according claim18 wherein the stimulation waveform optimized for energy efficiencyconsists essentially of a truncated Gaussian curve.
 22. A method forstimulating neurological tissue comprising applying a stimulationwaveform optimized for energy efficiency derived by a prescribed geneticalgorithm (GA) coupled to a computational model of extracellularstimulation of a mammalian myelinated axon.
 23. A method according toclaim 22 (i) wherein the genetic algorithm (GA) generates a populationof offspring stimulation waveforms by mating a population of parentstimulation waveforms, (ii) wherein the computational model ofextracellular stimulation of a mammalian myelinated axon assesses thefitness of individual offspring stimulation waveforms generated by theGA in terms of energy efficiency and selects as a new population ofparent stimulation waveforms the current offspring stimulation waveformshaving the highest energy efficiency values, and (iii) wherein (i) and(ii) are repeated until predetermined termination criteria is met.
 24. Amethod according to claim 23 wherein the termination criteria includesthe fitness of successive populations of offspring stimulation waveformsconverging toward a common energy-efficiency value.
 25. A methodaccording claim 23 wherein the stimulation waveform optimized for energyefficiency consists essentially of a Gaussian curve.
 26. A methodaccording claim 23 wherein the stimulation waveform optimized for energyefficiency consists essentially of a truncated Gaussian curve.
 27. Amethod for prolonging the battery life of an implantable pulse generatorcomprising conditioning the pulse generator to apply a stimulationwaveform optimized for energy efficiency derived by a prescribed geneticalgorithm (GA) coupled to a computational model of extracellularstimulation of a mammalian myelinated axon.
 28. A method according toclaim 27 (i) wherein the genetic algorithm (GA) generates a populationof offspring stimulation waveforms by mating a population of parentstimulation waveforms, (ii) wherein the computational model ofextracellular stimulation of a mammalian myelinated axon assesses thefitness of individual offspring stimulation waveforms generated by theGA in terms of energy efficiency and selects as a new population ofparent stimulation waveforms the current offspring stimulation waveformshaving the highest energy efficiency values, and (iii) wherein (i) and(ii) are repeated until predetermined termination criteria is met.
 29. Amethod according to claim 28 wherein the termination criteria includesthe fitness of successive populations of offspring stimulation waveformsconverging toward a common energy-efficiency value.
 30. A methodaccording claim 27 wherein the stimulation waveform optimized for energyefficiency consists essentially of a Gaussian curve.
 31. A methodaccording claim 28 wherein the stimulation waveform optimized for energyefficiency consists essentially of a truncated Gaussian curve.
 32. Amethod for generating an optimized waveform comprising coupling agenetic algorithm (GA) to a computational model of extracellularstimulation of a mammalian myelinated axon, and allowing the GA toprogress until the waveforms converge upon an energy-optimal shape,thereby generating the optimized waveform.
 33. A method according claim32 wherein the optimized waveform efficiency consists essentially of aGaussian curve.
 34. A method according claim 32 wherein the optimizedwaveform consists essentially of a truncated Gaussian curve.
 35. Amethod according to claim 32 wherein the optimized waveform ismonophasic.
 36. A method according to claim 32 wherein the optimizedwaveform is biphasic.